Scattering of dark matter and dark energy

Fergus Simpson*

SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill,Edinburgh EH9 3HJ, United Kingdom

(Received 2 August 2010; published 7 October 2010)

We demonstrate how the two dominant constituents of the Universe, dark energy and dark matter, could

possess a large scattering cross section without considerably impacting observations. Unlike interacting

models which invoke energy exchange between the two fluids, the background cosmology remains

unaltered, leaving fewer observational signatures. Following a brief review of the scattering cross sections

between cosmologically significant particles, we explore the implications of an elastic interaction between

dark matter and dark energy. The growth of large scale structure is suppressed, yet this effect is found to be

weak due to the persistently low dark energy density. Thus we conclude that the dark matter–dark energy

cross section may exceed the Thomson cross section by several orders of magnitude.

DOI: 10.1103/PhysRevD.82.083505 PACS numbers: 95.35.+d, 95.36.+x, 98.80.�k

I. INTRODUCTION

One of the most pressing issues in modern physicslies in the classification of dark energy. This is a phenomenonwhich not only appears to provide the bulk of the Universe’senergy content, but also gravitates in a repulsive manner,unlike any known substance. Prime candidates include thecosmological constant, scalar fields, and modifications toEinstein’s theory of gravity.

The first step in observationally distinguishing thesemodels involves studying the cosmic geometry, since thecosmological constant makes a strong predictive statementon the trajectory of the cosmic expansion. Over the pastdecade, progress in this area has seen the redshift-distancerelation tested by supernovae and baryon acoustic oscilla-tions (BAO) with a precision approaching �1%. Thistranslates into a bound on the dark energy equation of statew ’ �1� 0:1, where w � p=�.

However, studying the expansion history alone is insuf-ficient if we are to ever definitively exclude either scalarfields or modified theories of gravity. Therefore, it is also ofgreat importance to examine the growth of cosmic struc-ture, an area which is attracting growing attention. This canbe measured through various means such as redshift spacedistortions and weak gravitational lensing, though currentconstraints are relatively modest.

In performing this diagnosis of dark energy, we haveimplicitly been assuming that the physics within the darksector of cosmology—dark matter and dark energy—ispurely gravitational. Yet what limitations can we place ontheir physical behavior? While the precise nature of anymicrophysics is highly uncertain, the broader picture is onein which energy may be transferred either from dark energyto dark matter, or vice versa. Cosmologies with energyexchange have been extensively studied in the literature[1–10], and have been shown to leave characteristic

signatures within observables such as the integrated SachsWolfe effect, the Hubble constant H0, and the growth ofcosmic structure. Here we present a new class of modelswhich do not leave as clear a cosmological signal as thosewhich invoke energy exchange, since the comoving matterdensity remains fixed, and the background expansion isunaltered. Yet as we shall see, the growth of structure isreadily suppressed by a drag term arising from elasticscattering between the dark matter and dark energy fluids.

II. CROSS SECTIONS

In general, an interaction between two particles mayimpart a transfer of momentum or a transfer of energy, orlead to the creation of new particles. Which of these mightwe expect to arise from the dark matter–dark energy inter-action? Slow, low energy impacts (such as Thomson scat-tering and Rutherford scattering) often maintain elasticity,while relativistic velocities are more readily associated withinelastic events (such as Compton scattering and deepinelastic scattering). Given the extremely low dark energydensity, and the nonrelativistic velocities associated withdark matter motions, elastic scattering appears the simplestand most natural extension to dark sector physics. We neednot restrict ourselves to a particular physical model of darkenergy, as the results obtained are largely independent ofthe microphysics involved in the scattering process.The likelihood of scattering is quantified in terms of the

cross section, which may be thought of as the effectivetarget area as seen by an incident particle. We shall look toimpose an upper bound on the scattering cross section fordark matter–dark energy interactions, and place this withinthe context of other cross sections. Figure 1 reviews thescattering cross sections for a selection of cosmologicallysignificant particles, which we briefly review in the sub-sections below. Many of these interactions exhibit a strongenergy dependence, so in order to provide definitive valueswe adopt a cosmologically appropriate energy scale of*frgs@roe.ac.uk

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0.3 eV, corresponding to thermal collisions at the epoch ofrecombination (z� 1100).

One should also bear in mind that even with a fixedenergy scale, interactions may exhibit a significant depen-dence on other factors such as spin or environment. Thus,for simplicity, and to facilitate a visual comparison, wefocus on order of magnitude values.

A. Standard model scattering

The values of cross sections amongst standard modelparticles are generally well determined, albeit at muchhigher energy scales. The low energy values presented inFig. 1 are either based on theoretical prediction or a simpleextrapolation from higher energies.

(i) At low energies, the photon-electron interaction isgoverned by the Thomson cross section, �T ¼6:65� 10�25 cm2, which is of the order of 1 b.

(ii) Electron-electron scattering is divergent due to thelong range Coulomb interaction.

(iii) Photon-photon scattering is strongly suppressed atenergy scales below the electron rest mass, scalingas E6. This phenomenon has yet to be confirmedobservationally, although recent constraints areapproaching the required sensitivity [11].

(iv) At such low energies, neutrino-neutrino cross sec-tions are poorly understood; here we provide asimple extrapolation from higher energy scales [12].

(v) Neutrino-photon scattering is of astrophysicalimportance, as it is capable of significantly influenc-ing the evolution of stars and the dynamics of super-novae. However at sub-keVenergy scales, the elasticprocess dominates, leaving �ð�� ! ���Þ ��ð�� ! ��Þ [13].

(vi) Similarly, elastic scattering from neutral currentinteractions provides a prescription for theneutrino-electron value [14].

B. Dark-standard scattering

No direct detection of either dark matter or dark energyhas yet been made, so we are limited to applying upperbounds to these values. However, interactions betweenthe dark sector and standard model particles are quiterestricted.For quoted bounds involving dark matter, these scale

linearly as the particle mass, taken here to be 10 GeV=c2.(i) The dark matter-neutrino bound is based on the

detection of neutrinos arriving from SN1987A [15],which were not appreciably scattered by the inter-vening dark matter. Applying this analysis to theprojected dark energy density leaves a significantlymore modest constraint.

(ii) The Thomson optical depth established from obser-vations of the cosmic microwave background an-isotropies places a strong lower bound on the meanfree path of photons. This acts as a limit on theirinteractions with the dark sector.

(iii) If electrons were tightly coupled to dark matteror dark energy, this would impact on cosmicmicrowave background anisotropies.

C. Dark scattering

We are left with just three components.(i) The case of dark matter self-interactions has been

well studied [16–18]. This upper bound stems fromthe disruption of subhalos which would occur nearthe center of clusters. Note that bulk motions areunaffected, as only incoherent motions lead to scat-tering. This differs markedly from the case of darkmatter–dark energy scattering, which we explore indetail in the following section.

(ii) In order to maintain stable density perturbations,dark energy is required to have some internaldegrees of freedom. There may therefore be somelower bound on its self-interaction, but our ex-tremely limited understanding of dark energy phys-ics leaves dark energy–dark energy scattering themost uncertain component of Fig. 1.

(iii) Finally, the dark matter–dark energy cross sectionis the focus of this work. This weak bound isderived from the impact incurred on the growth oflarge scale structure, as outlined in the followingsection.

III. OBSERVATIONAL IMPACT

As discussed in the previous section, indicationsfrom known physics suggest that elastic scattering is themost abundant process at the energy scales of interest.

FIG. 1 (color online). A collection of cross sections betweencosmologically significant particles, in units of barns(10�24 cm2). We assume a collisional energy associated withthe era of recombination, 3000 K or equivalently �0:3 eV. Thedark matter particle is taken to have a mass of 10 GeV=c2, andthe dark energy equation of state w ¼ �0:9.

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Therefore, we shall explore a scenario in which dark matterscatters elastically within the dark energy fluid.

In order to define a cross section we must quantizedark energy in some manner. There are two regimes ofinterest—one in which the effective mass of dark energy ismuch greater or one in which it is much less than darkmatter. For instance, if a physical dark energy exists in a‘‘solid’’ configuration akin to a network of domain walls orcosmic strings [19], each dark matter particle experiences afinite mean free path before being subject to a dissipation-less recoil off the more massive structure. We shall explorethe ‘‘light’’ regime, treating the dark energy fluid as beingcomprised of relativistic particles, and assume the charac-teristic negative pressure arises via their self-interaction.In this toy model, the particles merely act as a proxy forthe energy density. However, ultimately our analysis of themacroscopic behavior and the conclusions drawn arelargely independent of the microphysics involved.

We begin by quantifying the impact dark scattering hason the growth of cosmic structure.

A. Large scale structure

The coupled differential equations governing the lineardensity and velocity perturbations �, � (see e.g. [3,20]) arenow modified, and we utilize the subscripts Q and c todenote the dark energy and dark matter fluids, respectively.Provided the dark energy quanta are light and relativistic(nonrelativistic particles would serve to increase the per-mitted cross section), the velocity perturbation exhibits anew drag term

�0Q ¼ 2H�Q � anD�D��þ k2�þ k2�Q

1þ w; (1)

where nD is the proper number density of dark matterparticles and �D the scattering cross section betweendark matter and dark energy, and we have defined thevelocity contrast �� � �Q � �c. The combination

nD�D�� represents the fraction of the dark energy quantawhich are subject to scattering per unit time. This is some-what analogous to the Thomson scattering term whichcouples baryons and photons. Conservation of momentumleads to a similar term arising in the equivalent equation fordark matter, and this introduces a dependence on the darkenergy equation of state.

�0c ¼ �H�c þ�Q

�c

ð1þ wÞanD�D��þ k2�; (2)

while the remaining perturbation equations are unchangedfrom their conventional form

�0Q ¼ �

�ð1þ wÞ þ 9

H2

k2ð1� w2Þ

��Q þ 3ð1þ wÞ�0

� 3Hð1� wÞ�Q; (3)

�0c ¼ ��c þ 3 _�: (4)

The Poisson equation provides the source term

k2� ¼ 4�Ga2Xi

�i�i; (5)

where we sum over the dark matter, dark energy, andbaryons. For our purposes the baryons are relatively inert,remaining unscattered by the dark energy fluid. The darkenergy sound speed is taken to be c2s ¼ 1. We work onscales sufficiently below the horizon, kH � 1, such thatthe dark energy’s large sound speed acts to maintain a highdegree of homogeneity.Previous studies of coupled dark energy models charac-

terize the energy-momentum transfer in terms of the four-vector Q, and those studies chose to align it with eitherthe dark energy or dark matter rest frames [3–7]. Herewe have effectively rotated Q to be spacelike, such thatQ0 ¼ 0. Since the comoving matter density is conserved,the background HðzÞ behaves no differently from that ofthe standard wCDM model.The evolution of density perturbations in Fig. 2 is

evaluated by numerical integration of the six coupleddifferential equations and is seen to depart significantlyfrom the zero-scattering model. The anomalous behaviorin the growth rate f � d ln�=d lna is more prevalent at latetimes, when there is simply more time available for inter-actions to occur. This fairly rapid onset of decelerationleads to the onset of baryon bias, with �b=�c ’ 1:1 at lowredshift. There are a number of potential tests for thisbaryon bias, from the composition of intracluster gas tothe motions of tidally disrupted stellar streams. It hasbeen noted that an apparent violation of the equivalence

FIG. 2 (color online). The logarithmic growth rate of lineardark matter perturbations, when subject to elastic scattering withthe dark energy fluid. For this configuration the particle massmD ¼ 10 GeV=c2, and w ¼ �0:9. The solid line corresponds toa cross section of �D ¼ 500 b, showing a suppression of growthat late times compared to the dotted line with no scattering(�D ¼ 0).

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principle of around 10% is the upper bound based oncurrent observations [21].

The modified growth history may also be interpreted interms of the growth index, defined such that

� � d lnf

d ln�m

: (6)

Ordinarily, general relativity predicts � ’ 6=11�15=2057�� [22], yet in Fig. 3 we see a significant depar-ture from this value, due to scattering between dark matterand dark energy. Provided w>�1, the drag term in (2)slows the growth of structure, enhancing the value of thegrowth index �. If one considers w<�1, the sign of thedrag term is reversed, thereby accelerating growth; how-ever, the physical interpretation of such a model is lessclear.

B. Redshift space distortions

One of the leading techniques for studying the growth oflarge scale structure is redshift space distortions. The ap-parent anisotropy of the galaxy power spectrum provides ameasure of the rate at which structure is forming on largescales. In Fig. 4 we demonstrate the rise in the growthindex as measured by a galaxy survey at z ¼ 0:5, combinedwith Planck, following the Fisher matrix prescriptionoutlined in [23]. This involves marginalizing over theparameter set

½w0; wa;��;�k;�mh2;�bh

2; ns; As; ; �; �p�: (7)

The standard cosmological parameters are taken to havefiducial values as derived from the five-year WilkinsonMicrowave Anisotropy Probe observations [24]. The con-tours plotted represent the estimated 1- and 2-� likelihoodcontours.The solid line corresponds to a cross section �D ¼

300 b and equation of state w ¼ �0:9. Unlike energyexchange models, � is the only cosmological parametersubject to a bias. The dashed contours correspond to theenergy exchange model outlined in [7], where dark energydecays into a form of dark matter.

C. Virialized structures

On smaller scales, consider a dark matter halo atrest in the dark energy frame. Elastic scattering acts in asimilar manner to dark matter self-scattering, which wouldtend to isotropize the halo. However, if we introduce avelocity-dependent cross section, halos with a large pecu-liar velocity could exhibit an unusual behavior. Dark mat-ter particles with motions aligned with the peculiarvelocity would be subject to a greater retardation force,and this may influence the orientation of the halo’sellipticity. A correlation of halo alignment with peculiarvelocity may therefore be indicative of interactions in thedark sector.

IV. DISCUSSION

In the event that dark energy takes some physical form(neither a cosmological constant nor a manifestation ofnew gravitational physics), then we might expect it to

FIG. 3 (color online). The evolution in the growth index as afunction of the scale factor. Thick solid and dashed lines corre-spond to models of dark energy with w ¼ �0:9 and w ¼ �0:99,respectively. As with Fig. 2, the dark matter–dark energy crosssection is taken to be 500 b. The dotted line representsthe standard case of zero scattering. Below the dotted line,the thin solid and dashed lines correspond to w ¼ �1:1 andw ¼ �1:01 models.

FIG. 4 (color online). The solid contours demonstrate themodification to the growth index induced by dark matter–darkenergy scattering, with the cross section taken to be �D ¼ 300 b.The dashed contours provide an example of the bias which maybe induced in the gravitational growth index � by the interactingmodel outlined in [7]. The standard model is indicated by theblack dot.

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interact in some additional nongravitational manner. Of theinteraction cross sections in Fig. 1 which are known, theyare all nonzero and predominantly elastic. One mightimagine that any such coupling within the dark sectormust be extremely weak, in order to allow dark matterparticles to experience a very long mean free path.However, owing to the persistently low energy density ofdark energy, and the fairly low number density of darkmatter particles, quite considerable cross sections are per-mitted. For an equation of state w ¼ �0:9, this can exceedthe Thomson cross section by 2 orders of magnitude beforea significant impact is made on the growth of large scalestructure. As we approach the limitw ¼ �1, our constraintweakens further.

Of course there are many subtleties which could alter theform of the interaction, such as a velocity-dependent crosssection. This model simply provides a demonstration of thevast volume of parameter space available for interactions

between dark matter and dark energy to persist and evadedetection.In this class of models, we have identified a modification

to the growth rate, and an induced baryon bias, two featureswhich are also associated with energy exchange. However,unlike models with energy exchange, the comoving matterdensity is conserved, and the expansion history remainsunperturbed. In addition, a characteristic signature mayreside in the alignment of dark matter halos with theirdirection of motion.

ACKNOWLEDGMENTS

The author would like to thank Brendan Jackson, AndyTaylor, John Peacock, Pedro Ferreira, ConstantinosSkordis, and Ed Copeland for helpful discussions and theanonymous referee for useful feedback, and acknowledgesthe support of an STFC rolling grant.

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